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A classification of matrices of class Z
A Classification
of Matrices of Class Z
Miroslav Fiedler
Mathemutics Zn-stitute CzechosZovak Academy of Sciences &tnri 25, Prahu 1, 11567 Czechoslovakia and Thomas L. Markham*
Department of Mathemutics University of South Carolina Columbia, South Carolina 29208
Submittedby Hans Schneider
ABSTRACT
We generalize the classes N,, and F. studied by K. Fan, G. Johnson, and R. Smith. Schur complements and lattices are examined for matrices in these classes.
1.
A GENERALIZATION
OF A’, AND F’, MATRICES
Suppose A is a matrix over the field of real numbers. Throughout,
we deal
with N x N Z-matrices, which are matrices whose off-diagonal entries are nonpositive. We will use the notation of Fiedler and Ptik [3] with regard to class K and class K,, which are respectively the class of nonsingular Mmatrices and the class of M-matrices. Ky Fan [2] defined N-matrices to have the form
A=
tZ-
B,
(1)
*The work of this author was done while visiting at the Mathematics LZNEAR ALGEBRA
AND ITS APPLZCATZONS 173:115-124
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
Institute in Prague. 115
(1992)
0024-3795/92/$5.00
116
MIROSLAV
where B of B and principal negative
FIEDLER
AND THOMAS
L. MARKHAM
> 0, and p,_ 1( B) < t < p(B), where p(B) is the Perron eigenvalue pn _ 1( B) is the maximum of the spectral radii of the (12 - 1) X (n - 1) submatrices of B. He showed that N-matrices are Z-matrices with determinant and with proper principal submatrices belonging to K.
G. Johnson [5] extended Fan’s definition to No-matrices. He required that No-matrices have the form given in (1) with P,_~( LS) < t < p(B). He also studied matrices of the form (1) for n > 3, where P,_~(B) < t < P,_~(B), where pn_e is the maximum spectral radius of the (n - 2) x (n - 2) principal submatrices of B. Smith [S] called these matrices F,-matrices, in honor of Fan. A further step was taken by Ying Chen [l], who also studied inverse Fo-matrices and inverse N,,-matrices. We intend to generalize
these ideas in the following manner.
DEFINITION 1.1. Let L, denote the class of real n x n matrices have the form A = tZ - B where 23 2 0 and
which
where
s), for
p,(B)
= max{ p( g) : g is a principal
submatrix
of B of order
s = 1, 2, . . . , n. If A EL,,
then we say the height of A in Z is s.
Using our notation, it is clear that L,_, = No, L,_, = F,, L, = Ko. For convenience, we let p,( B)p,+,(B) = - 00 for s = 0, p + l(B) = 03 fir s = n. None of the classes Lo, L,, . . . , L, are void, as the following example demonstrates. EXAMPLE 1. Let J be the n x n matrix of all ones. Clearly, pk(_/) = k for k = 1,2,. . . , n.Iftsatisfiesk
If B 2 0, it is well known that
If B is strictly positive, the inequalities are strict. Further,
if B is irreducible,
then p,_,(B) < P,(B) = P(B). Too, we note that the class L, is invariant under permutational similarity. The classes Lo, L,, . . . , L, form a decomposition of the class Z in the sense that if A E Z, then A belongs to exactly one of these classes. This means, of course, that the height of A is well defined.
CLASSIFICATION
117
OF MATRICES OF CLASS Z
If A is a triangular
matrix,
then
A E I,,
if all diagonal entries
of A are
nonnegative. If at least one diagonal entry of A is negative, then A belongs to La. Thus, the triangular matrices belong only to the extreme classes. The same is true also for diagonal matrices and for matrices obtained from triangular matrices by simultaneous permutations of rows and columns. Next, we give an alternative definition of L,, which is sometimes useful in our work. I 1 < s < n - 1. Let L, denote the class of DEFINITION 1.2. Suppose n x n Z-matrices which have the property that if A E Z, all principal submatrices of A of order s belong to K,, but there exists a principal submatrix order s + 1 of A which is not in K,. We define &, = L, and i,, = L,. THEOREM1.3. Prooj. Lets=1 If AEL, then
The classes L, and i, ,...,
are identical for s = 0, . . , n. n
n-l.
A = tZ - B, where
Clearly, A E L,. If A EL,, then since
A E Z, we write
scalar t. Since all principal
of
submatrices
B > 0
and
p,(B)
< t
A = tl - B with B 2 0 and some of A of order s are in Ku, then we
must have t 2 p,(B). If t 2 ps+i( B), tb en all submatrices of A of order s + 1 would be in K,, which is a contradiction. Thus p,(B) < t < P,+~( B), and A is in L,. COROLLARY1.4. Zf A E L, and D is an n x n positive diagonal matrix, then DA and AD belong to L, for s = 0, 1, . . . , n. Proof
The proof follows immediately
from Theorem
1.3, since a positive
diagonal scaling of a matrix in Kc remains in K,, and if a principal submatrix of order s + 1 has a negative determinant, then a positive diagonal scaling of this submatrix also has a negative determinant. to 2,.
Thus both DA and AD belong H
If A and B are n x n real matrices, we write as usual A < B whenever aij < b, for all pairs (i, j). The following is a generalization of G. Johnson’s Theorem 2.10(i) class L,.
in [5], and also shows the interplay
THEOREM1.5.
L~AEL,,
BEL,
withA
of monotonicity
with the
Then
(i) s < t, and (ii) whenever a matrix C satisfies A < C < B, then C E L,, where s < 9 < t.
118
MIROSLAV FIEDLER AND THOMAS L. MARKHAM
Proof. It suffices to prove the second part only. Since C E Z, C belongs to some L,, 0 < v < n. There exist nonnegative
matrices
P, Q, and R such that
A = XI - P,
PS( P) G x < &+1(P)?
B = hI - Q.
P,(Q) G h
c= AZ- R,
P”(R) G X < P,+I( fl).
Since A < C < B, the matrices
(4)
P, Q, R satisfy
By a well-known property of nonnegative
P,(P) 2 P”(R)2
matrices,
u=O,...,n.
P,(Q),
But then it follows from (4) that
LEMMA 1.6.
indicess, Proof.
t,s
ZfA = hZ - Bandp,(B)
= P,+~(B) = *** = p,(B)forsome thenAcannotbelongtoL,firs
Immediate.
THEOREM 1.7. Let A EL, be reducible and s < n. Write A = XI - P, where p > 0 has diagonal blocks (in the Frobenius fm) P,, . . . , Pk. Zf p(P) = p( P$, then the order of q is greater than s.
Proof.
Let v be the order of pi. Since p,(P) 3 p( pj), we have, by (3), P”(P)
= P”,l(P)
Since s < n, it follows from Lemma COROLLARY1.8 (G. Johnson [5]).
=
***
=
P”(P).
1.6 that v > s. All matrices in L, _ 1 are irreducible.
n
CLASSIFICATION OF MATRICES OF CLASS Z EXAMPLE 2.
In the theorem,
one cannot
the following example will illustrate.
where (a) A,, then
expect
a stronger
assertion,
as
If
[lnikA:,].
A=
Lk_i,
119
is k x k, (b) A,, = Z - P, where P > 0, and (c) A,, belongs to
A also belongs to L&i.
We shall also need the following example. EXAMPLE 3. In Theorem 1.7, one cannot expect a stronger inequality the order of Pj, as we now illustrate.
for
Let Jr,, denote the r x t matrix of all ones. If r = t, we write Jr for simplicity. Now suppose that s and t are numbers such that 1 < s < t < n. Let
0
&.I%-3 0 0 where
E > 0 is chosen
small.
Let
1
EJt--s,n--t T 0
A = XI - B. Then
it is easily seen that
P,._~ = s - 1 but s = p,(B) = p,+,(B) = . ** = p,(B) < P~+~(B). Therefore, the matrix XI - B belongs to L, for s - 1 < X < s, but SI - B E L,. In the sequel, we shall investigate First, we denote by aL, obtained in the left-hand-side
aL,=
(AEZ:
further properties
the set of matrices inequality in (2):
A=hl-
We call aL, the lower boundary
B,X=p,(B)),
of the classes
L,.
in Z for which equality
s=
is
l,...,n.
of L,.
It is clear that all the sets L$aL,, s = 1, . . . , n - 1, are open sets in Z in the sense that whenever A E LsUL,, there exists a neighborhood N of A such that every matrix in N n Z belongs to L,\aL, as well. Also, L, is open. We can ask also about the closure ES of L,, i.e., the set consisting of all the limits of all convergent sequences of matrices belonging to L,. It is clear that a matrix in Lj with j < s cannot be in ES. Also, a matrix in Lt\i3L, cannot belong to L, for t > s. The following theorem shows that all other cases can occur.
120
MIROSLAV THEOREM 1.9.
FIEDLER
AND THOMAS
L. MARKHAM
Let p, q be integers satisfying 0 < p < q < n. Then
Z,naL,+rz.. REMARK. Due to the previous observation, Proof. We shall distinguish five cases. matrix V E a L, and a matrix U, and set
&, n aL,
= zr, n L,.
In each case, we shall specify a
1
Ak = -u+v,
k=
k+l
Thus V = limk,, Case 1.
Case 2.
1,2,....
Ak, and we always have that A, EL,.
p = 0, q = n. p = 0, q < n.
We set V = 0, 1. = -I. Define V as
Clearly A, EL,.
n-q q- 1, 1
v=
where e is the vector of all ones, and U = -I. Case 3. p = 1, q Q n. Set V as in case 2, U = Z - J, where J is a matrix of all ones. Set V= (p+ 1)1-J, U= -1. Case4. p>2,q=p+l.
Case 5. ~22, p+l
2.
THE SCHUR
n
COMPLEMENT
Suppose A is an n x n matrix partitioned
as
If A,, is a square submatrix of A which is invertible, the Schur complement A,, in A is A/Air:=
A22
-
A21
All%2.
of
(6)
CLASSIFICATION If
A belongs
121
OF MATRICES OF CLASS 2
to L,,
we will say that
A is pure in L, if every principal
submatrix of order s + 1 is negative. We can now prove the following theorem.
Suppose A is a matrix in L,, and A,, is a k x k invertible THEOREM 2.1. submatrix of A, as in (5), where k < s. (i) Then A/A,, E Lj for some j 2 s - k. (ii) If A is pure in L,, then A/A,, belongs to L,_,.
Proof. First, if A has order n, then A/A,, has order n - k. Let us index the rows and columns by k + 1,. . . , n. If OL is a sequence chosen from , R, we let A/A,,(a) denote the principal submatrix of A /A,, k+ l,... whose rows and columns are indexed by 01. First, we note that A /A,, E 2, since A E Z and A,’
2 0. It is well known
that
det A/A,,((r,,
det A(l,.
. . . , ak) =
. . , k, ccl,. . . , ak) det A,,
(7)
Hence all principal minors of A /A, r of order ,< s - k are nonnegative. This proves the first assertion. If A is pure, there exists a principal minor of A/A,, of order s - k + 1 which is negative.
EXAMPLE 4.
By Theorem
1.1, A EL,_,.
We return to the matrix of Example
X > s, the Schur complement
n
3. For A = Xl - B and
A/XI - jS has the form
which clearly belongs to L,_,. Although A EL, where t > s, we have A /A,, EL,_, for n - s > n - t. This shows that in the nonpure case, (i) cannot be improved.
122
3.
MIROSLAV
LATTICES
ASSIGNED
FIEDLER
TO A SQUARE
AND THOMAS
L. MARKHAM
MATRIX
In this section, we shall investigate the sign properties of the principal minors of a matrix in Z. For better understanding of these properties, we now introduce two definitions and some notation.
DEFINITION 3.1.
Let
A be a real
n x n matrix.
By the lattice
of A,
denoted by U(A), we mean the directed graph with 2” points, each of which corresponds to exactly one subset M of the index set N = (1, 2,. . . , n} and is assigned the sign + 1, 0, or - 1 of the principal minor det A(M) if M + 0; the sign associated with 0 is + 1; the edges of U(A) are the ordered pairs (M,, Ma) for which M, c M, and 1 M, 1 = 1 M, 1 + 1. We also call 1M I the level of the point M. It is clear that the lattice of a matrix does not essentially change if we permute the rows and columns simultaneously. We shall denote by T the point of the lattice corresponding and B will be the point corresponding to the void set. In the sequel, we denote by 2, the class of n x n matrices principal submatrices of all orders are nonsingular.
to the set N, in Z whose
DEFINITION 3.2. Let A E 2,. By the signed lattice of A, denoted by _!?(A), we mean the lattice Y(A) defined earlier to which each edge (i, k) is assigned a sign + 1 or - 1 according
as i and k have the same sign or opposite
signs.
THEOREM3.1.
Let A E Z. Then
U( A) has the following
properties:
(i) i’f P is a point in U( A) such that one path from P to B consists of positively signed points only, then every path from P to B in U(A) has this property. (ii) The height of A is equal either to n, or to the smallest point (if such exists) diminished by one.
level of a negative
Proof. The first property follows from the well-known property of matrices of class Z (cf. [3, Theorem 4.31) that if there is one nested sequence of positive principal minors having length equal to the order of the matrix, then the matrix belongs to K and hence all nested sequences of principal minors of this length consist of positive elements. n The second assertion follows from the definitions.
CLASSIFICATION OF MATRICES OF CLASS 2 THEOREM3.2. (i)
AII paths
number
of positive
LRt A be a symmetric in @(A)
matrix
123 in 2,.
f ram the point T to the point B contain of course, of negatiue edges).
the same
edges (and,
(ii) lf A(M) is a principal s&matrix of A belonging to the class K, and M c N with ) M ) = s, then_ the Schur compkment A/ A(M) is a mat+ in Z, _ 2, and its signed lattice Y ( A / A( M 1) is obtained as the sublattice of Y(A) induced (together with the signs) by U( A) on the set of those points which correspond to the subsets of indices of M.
Proof.
The first assertion follows by the Jacobi criterion:
In(A), is (rr, Y, 0), where one, and Y is the number
the inertia of A,
r is the number of coincidences of the signs plus of changes of the signs in one nested sequence of
principal minors of A of length n (cf. [4, p. 2721). The second assertion follows immediately from the formula (7).
n
THEOREM 3.3. Let A be an n x n matrix in Z with height n (and thus in the class Kc). Then if Y ( A) contains a point M with sign zero, then every point from which there exists a path to M in Y( A) has also sign zero. Proof. This follows from the fact (cf. [3, Theorems 5.5, 4.31) that if A,, is a singular submatrix of A E K,, then every principal submatrix of A containing
n
A,, is again singular. The following notion could have importance DEFINITION 3.3.
Let
for matrices
in class 2.
A E 2. We say that A has positivity length
the order of a maximal principal
submatrix
belonging
1 if 1 is
to K,.
REMARK. For U(A), 1 is the length of the longest nonnegative path in Y(A) ending in B. Clearly, the matrices contained in proper classes L, are characterized by the fact that their positivity length coincides with their height.
4.
CONCLUDING
REMARKS
CONJECTURE. If Y is a lattice with subsets of N= (I,... , n}, and the signs and satisfy condition (i) of Theorem 3.1 then there exists an n x n matrix A E K,
2” points corresponding to all the of the points belong only to (0, 11 and the condition of Theorem 3.3, such that _I?= Y(A).
124
MIROSLAV
FIEDLER
AND THOMAS
L. MARKHAM
The class of Z-matrices is closely related to the class of essentially nonnegative matrices of Varga [7]. Indeed, A E Z if and only if -A is essentially nonnegative. Thus, all definitions and results of this paper have their counterpart for essentially nonnegative matrices. REFERENCES 1
Ying Chen, (1991).
2
Ky Fan, Some matrix inequalities, Abh. Math. Sem. Univ. Hamburg 29:185-196 (1966). M. Fiedler and V. Ptak, On matrices with nonpositive off-diagonal ele-
3
Notes
on FO-matrices,
ments and positive principal (1962). 4 5
minors,
Linear
Algebra
Czechoslovak
Appl.
Math.
142:167-172
J. 42:382-400
F. R. Gantmacher, Theory of Matrices, Nauka, Moscow, 1953. G. Johnson, A generalization of N-matrices, Linear Algebra 48:201-217 (1982).
6
R. L. Smith, Bounds on the spectrum
7
Z-matrices, Linear Algebra Appl., 129:13-28 (1990). R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, N.J., 1962. Received 27 December 1990; final
nzanuscript
of nonnegative
matrices
accepted 9 July 1991
Appl.
and certain
Englewood
Cliffs,
of Matrices of Class Z
Miroslav Fiedler
Mathemutics Zn-stitute CzechosZovak Academy of Sciences &tnri 25, Prahu 1, 11567 Czechoslovakia and Thomas L. Markham*
Department of Mathemutics University of South Carolina Columbia, South Carolina 29208
Submittedby Hans Schneider
ABSTRACT
We generalize the classes N,, and F. studied by K. Fan, G. Johnson, and R. Smith. Schur complements and lattices are examined for matrices in these classes.
1.
A GENERALIZATION
OF A’, AND F’, MATRICES
Suppose A is a matrix over the field of real numbers. Throughout,
we deal
with N x N Z-matrices, which are matrices whose off-diagonal entries are nonpositive. We will use the notation of Fiedler and Ptik [3] with regard to class K and class K,, which are respectively the class of nonsingular Mmatrices and the class of M-matrices. Ky Fan [2] defined N-matrices to have the form
A=
tZ-
B,
(1)
*The work of this author was done while visiting at the Mathematics LZNEAR ALGEBRA
AND ITS APPLZCATZONS 173:115-124
0 Elsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010
Institute in Prague. 115
(1992)
0024-3795/92/$5.00
116
MIROSLAV
where B of B and principal negative
FIEDLER
AND THOMAS
L. MARKHAM
> 0, and p,_ 1( B) < t < p(B), where p(B) is the Perron eigenvalue pn _ 1( B) is the maximum of the spectral radii of the (12 - 1) X (n - 1) submatrices of B. He showed that N-matrices are Z-matrices with determinant and with proper principal submatrices belonging to K.
G. Johnson [5] extended Fan’s definition to No-matrices. He required that No-matrices have the form given in (1) with P,_~( LS) < t < p(B). He also studied matrices of the form (1) for n > 3, where P,_~(B) < t < P,_~(B), where pn_e is the maximum spectral radius of the (n - 2) x (n - 2) principal submatrices of B. Smith [S] called these matrices F,-matrices, in honor of Fan. A further step was taken by Ying Chen [l], who also studied inverse Fo-matrices and inverse N,,-matrices. We intend to generalize
these ideas in the following manner.
DEFINITION 1.1. Let L, denote the class of real n x n matrices have the form A = tZ - B where 23 2 0 and
which
where
s), for
p,(B)
= max{ p( g) : g is a principal
submatrix
of B of order
s = 1, 2, . . . , n. If A EL,,
then we say the height of A in Z is s.
Using our notation, it is clear that L,_, = No, L,_, = F,, L, = Ko. For convenience, we let p,( B)p,+,(B) = - 00 for s = 0, p + l(B) = 03 fir s = n. None of the classes Lo, L,, . . . , L, are void, as the following example demonstrates. EXAMPLE 1. Let J be the n x n matrix of all ones. Clearly, pk(_/) = k for k = 1,2,. . . , n.Iftsatisfiesk
If B 2 0, it is well known that
If B is strictly positive, the inequalities are strict. Further,
if B is irreducible,
then p,_,(B) < P,(B) = P(B). Too, we note that the class L, is invariant under permutational similarity. The classes Lo, L,, . . . , L, form a decomposition of the class Z in the sense that if A E Z, then A belongs to exactly one of these classes. This means, of course, that the height of A is well defined.
CLASSIFICATION
117
OF MATRICES OF CLASS Z
If A is a triangular
matrix,
then
A E I,,
if all diagonal entries
of A are
nonnegative. If at least one diagonal entry of A is negative, then A belongs to La. Thus, the triangular matrices belong only to the extreme classes. The same is true also for diagonal matrices and for matrices obtained from triangular matrices by simultaneous permutations of rows and columns. Next, we give an alternative definition of L,, which is sometimes useful in our work. I 1 < s < n - 1. Let L, denote the class of DEFINITION 1.2. Suppose n x n Z-matrices which have the property that if A E Z, all principal submatrices of A of order s belong to K,, but there exists a principal submatrix order s + 1 of A which is not in K,. We define &, = L, and i,, = L,. THEOREM1.3. Prooj. Lets=1 If AEL, then
The classes L, and i, ,...,
are identical for s = 0, . . , n. n
n-l.
A = tZ - B, where
Clearly, A E L,. If A EL,, then since
A E Z, we write
scalar t. Since all principal
of
submatrices
B > 0
and
p,(B)
< t
A = tl - B with B 2 0 and some of A of order s are in Ku, then we
must have t 2 p,(B). If t 2 ps+i( B), tb en all submatrices of A of order s + 1 would be in K,, which is a contradiction. Thus p,(B) < t < P,+~( B), and A is in L,. COROLLARY1.4. Zf A E L, and D is an n x n positive diagonal matrix, then DA and AD belong to L, for s = 0, 1, . . . , n. Proof
The proof follows immediately
from Theorem
1.3, since a positive
diagonal scaling of a matrix in Kc remains in K,, and if a principal submatrix of order s + 1 has a negative determinant, then a positive diagonal scaling of this submatrix also has a negative determinant. to 2,.
Thus both DA and AD belong H
If A and B are n x n real matrices, we write as usual A < B whenever aij < b, for all pairs (i, j). The following is a generalization of G. Johnson’s Theorem 2.10(i) class L,.
in [5], and also shows the interplay
THEOREM1.5.
L~AEL,,
BEL,
withA
of monotonicity
with the
Then
(i) s < t, and (ii) whenever a matrix C satisfies A < C < B, then C E L,, where s < 9 < t.
118
MIROSLAV FIEDLER AND THOMAS L. MARKHAM
Proof. It suffices to prove the second part only. Since C E Z, C belongs to some L,, 0 < v < n. There exist nonnegative
matrices
P, Q, and R such that
A = XI - P,
PS( P) G x < &+1(P)?
B = hI - Q.
P,(Q) G h
c= AZ- R,
P”(R) G X < P,+I( fl).
Since A < C < B, the matrices
(4)
P, Q, R satisfy
By a well-known property of nonnegative
P,(P) 2 P”(R)2
matrices,
u=O,...,n.
P,(Q),
But then it follows from (4) that
LEMMA 1.6.
indicess, Proof.
t,s
ZfA = hZ - Bandp,(B)
= P,+~(B) = *** = p,(B)forsome thenAcannotbelongtoL,firs
Immediate.
THEOREM 1.7. Let A EL, be reducible and s < n. Write A = XI - P, where p > 0 has diagonal blocks (in the Frobenius fm) P,, . . . , Pk. Zf p(P) = p( P$, then the order of q is greater than s.
Proof.
Let v be the order of pi. Since p,(P) 3 p( pj), we have, by (3), P”(P)
= P”,l(P)
Since s < n, it follows from Lemma COROLLARY1.8 (G. Johnson [5]).
=
***
=
P”(P).
1.6 that v > s. All matrices in L, _ 1 are irreducible.
n
CLASSIFICATION OF MATRICES OF CLASS Z EXAMPLE 2.
In the theorem,
one cannot
the following example will illustrate.
where (a) A,, then
expect
a stronger
assertion,
as
If
[lnikA:,].
A=
Lk_i,
119
is k x k, (b) A,, = Z - P, where P > 0, and (c) A,, belongs to
A also belongs to L&i.
We shall also need the following example. EXAMPLE 3. In Theorem 1.7, one cannot expect a stronger inequality the order of Pj, as we now illustrate.
for
Let Jr,, denote the r x t matrix of all ones. If r = t, we write Jr for simplicity. Now suppose that s and t are numbers such that 1 < s < t < n. Let
0
&.I%-3 0 0 where
E > 0 is chosen
small.
Let
1
EJt--s,n--t T 0
A = XI - B. Then
it is easily seen that
P,._~ = s - 1 but s = p,(B) = p,+,(B) = . ** = p,(B) < P~+~(B). Therefore, the matrix XI - B belongs to L, for s - 1 < X < s, but SI - B E L,. In the sequel, we shall investigate First, we denote by aL, obtained in the left-hand-side
aL,=
(AEZ:
further properties
the set of matrices inequality in (2):
A=hl-
We call aL, the lower boundary
B,X=p,(B)),
of the classes
L,.
in Z for which equality
s=
is
l,...,n.
of L,.
It is clear that all the sets L$aL,, s = 1, . . . , n - 1, are open sets in Z in the sense that whenever A E LsUL,, there exists a neighborhood N of A such that every matrix in N n Z belongs to L,\aL, as well. Also, L, is open. We can ask also about the closure ES of L,, i.e., the set consisting of all the limits of all convergent sequences of matrices belonging to L,. It is clear that a matrix in Lj with j < s cannot be in ES. Also, a matrix in Lt\i3L, cannot belong to L, for t > s. The following theorem shows that all other cases can occur.
120
MIROSLAV THEOREM 1.9.
FIEDLER
AND THOMAS
L. MARKHAM
Let p, q be integers satisfying 0 < p < q < n. Then
Z,naL,+rz.. REMARK. Due to the previous observation, Proof. We shall distinguish five cases. matrix V E a L, and a matrix U, and set
&, n aL,
= zr, n L,.
In each case, we shall specify a
1
Ak = -u+v,
k=
k+l
Thus V = limk,, Case 1.
Case 2.
1,2,....
Ak, and we always have that A, EL,.
p = 0, q = n. p = 0, q < n.
We set V = 0, 1. = -I. Define V as
Clearly A, EL,.
n-q q- 1, 1
v=
where e is the vector of all ones, and U = -I. Case 3. p = 1, q Q n. Set V as in case 2, U = Z - J, where J is a matrix of all ones. Set V= (p+ 1)1-J, U= -1. Case4. p>2,q=p+l.
Case 5. ~22, p+l
2.
THE SCHUR
n
COMPLEMENT
Suppose A is an n x n matrix partitioned
as
If A,, is a square submatrix of A which is invertible, the Schur complement A,, in A is A/Air:=
A22
-
A21
All%2.
of
(6)
CLASSIFICATION If
A belongs
121
OF MATRICES OF CLASS 2
to L,,
we will say that
A is pure in L, if every principal
submatrix of order s + 1 is negative. We can now prove the following theorem.
Suppose A is a matrix in L,, and A,, is a k x k invertible THEOREM 2.1. submatrix of A, as in (5), where k < s. (i) Then A/A,, E Lj for some j 2 s - k. (ii) If A is pure in L,, then A/A,, belongs to L,_,.
Proof. First, if A has order n, then A/A,, has order n - k. Let us index the rows and columns by k + 1,. . . , n. If OL is a sequence chosen from , R, we let A/A,,(a) denote the principal submatrix of A /A,, k+ l,... whose rows and columns are indexed by 01. First, we note that A /A,, E 2, since A E Z and A,’
2 0. It is well known
that
det A/A,,((r,,
det A(l,.
. . . , ak) =
. . , k, ccl,. . . , ak) det A,,
(7)
Hence all principal minors of A /A, r of order ,< s - k are nonnegative. This proves the first assertion. If A is pure, there exists a principal minor of A/A,, of order s - k + 1 which is negative.
EXAMPLE 4.
By Theorem
1.1, A EL,_,.
We return to the matrix of Example
X > s, the Schur complement
n
3. For A = Xl - B and
A/XI - jS has the form
which clearly belongs to L,_,. Although A EL, where t > s, we have A /A,, EL,_, for n - s > n - t. This shows that in the nonpure case, (i) cannot be improved.
122
3.
MIROSLAV
LATTICES
ASSIGNED
FIEDLER
TO A SQUARE
AND THOMAS
L. MARKHAM
MATRIX
In this section, we shall investigate the sign properties of the principal minors of a matrix in Z. For better understanding of these properties, we now introduce two definitions and some notation.
DEFINITION 3.1.
Let
A be a real
n x n matrix.
By the lattice
of A,
denoted by U(A), we mean the directed graph with 2” points, each of which corresponds to exactly one subset M of the index set N = (1, 2,. . . , n} and is assigned the sign + 1, 0, or - 1 of the principal minor det A(M) if M + 0; the sign associated with 0 is + 1; the edges of U(A) are the ordered pairs (M,, Ma) for which M, c M, and 1 M, 1 = 1 M, 1 + 1. We also call 1M I the level of the point M. It is clear that the lattice of a matrix does not essentially change if we permute the rows and columns simultaneously. We shall denote by T the point of the lattice corresponding and B will be the point corresponding to the void set. In the sequel, we denote by 2, the class of n x n matrices principal submatrices of all orders are nonsingular.
to the set N, in Z whose
DEFINITION 3.2. Let A E 2,. By the signed lattice of A, denoted by _!?(A), we mean the lattice Y(A) defined earlier to which each edge (i, k) is assigned a sign + 1 or - 1 according
as i and k have the same sign or opposite
signs.
THEOREM3.1.
Let A E Z. Then
U( A) has the following
properties:
(i) i’f P is a point in U( A) such that one path from P to B consists of positively signed points only, then every path from P to B in U(A) has this property. (ii) The height of A is equal either to n, or to the smallest point (if such exists) diminished by one.
level of a negative
Proof. The first property follows from the well-known property of matrices of class Z (cf. [3, Theorem 4.31) that if there is one nested sequence of positive principal minors having length equal to the order of the matrix, then the matrix belongs to K and hence all nested sequences of principal minors of this length consist of positive elements. n The second assertion follows from the definitions.
CLASSIFICATION OF MATRICES OF CLASS 2 THEOREM3.2. (i)
AII paths
number
of positive
LRt A be a symmetric in @(A)
matrix
123 in 2,.
f ram the point T to the point B contain of course, of negatiue edges).
the same
edges (and,
(ii) lf A(M) is a principal s&matrix of A belonging to the class K, and M c N with ) M ) = s, then_ the Schur compkment A/ A(M) is a mat+ in Z, _ 2, and its signed lattice Y ( A / A( M 1) is obtained as the sublattice of Y(A) induced (together with the signs) by U( A) on the set of those points which correspond to the subsets of indices of M.
Proof.
The first assertion follows by the Jacobi criterion:
In(A), is (rr, Y, 0), where one, and Y is the number
the inertia of A,
r is the number of coincidences of the signs plus of changes of the signs in one nested sequence of
principal minors of A of length n (cf. [4, p. 2721). The second assertion follows immediately from the formula (7).
n
THEOREM 3.3. Let A be an n x n matrix in Z with height n (and thus in the class Kc). Then if Y ( A) contains a point M with sign zero, then every point from which there exists a path to M in Y( A) has also sign zero. Proof. This follows from the fact (cf. [3, Theorems 5.5, 4.31) that if A,, is a singular submatrix of A E K,, then every principal submatrix of A containing
n
A,, is again singular. The following notion could have importance DEFINITION 3.3.
Let
for matrices
in class 2.
A E 2. We say that A has positivity length
the order of a maximal principal
submatrix
belonging
1 if 1 is
to K,.
REMARK. For U(A), 1 is the length of the longest nonnegative path in Y(A) ending in B. Clearly, the matrices contained in proper classes L, are characterized by the fact that their positivity length coincides with their height.
4.
CONCLUDING
REMARKS
CONJECTURE. If Y is a lattice with subsets of N= (I,... , n}, and the signs and satisfy condition (i) of Theorem 3.1 then there exists an n x n matrix A E K,
2” points corresponding to all the of the points belong only to (0, 11 and the condition of Theorem 3.3, such that _I?= Y(A).
124
MIROSLAV
FIEDLER
AND THOMAS
L. MARKHAM
The class of Z-matrices is closely related to the class of essentially nonnegative matrices of Varga [7]. Indeed, A E Z if and only if -A is essentially nonnegative. Thus, all definitions and results of this paper have their counterpart for essentially nonnegative matrices. REFERENCES 1
Ying Chen, (1991).
2
Ky Fan, Some matrix inequalities, Abh. Math. Sem. Univ. Hamburg 29:185-196 (1966). M. Fiedler and V. Ptak, On matrices with nonpositive off-diagonal ele-
3
Notes
on FO-matrices,
ments and positive principal (1962). 4 5
minors,
Linear
Algebra
Czechoslovak
Appl.
Math.
142:167-172
J. 42:382-400
F. R. Gantmacher, Theory of Matrices, Nauka, Moscow, 1953. G. Johnson, A generalization of N-matrices, Linear Algebra 48:201-217 (1982).
6
R. L. Smith, Bounds on the spectrum
7
Z-matrices, Linear Algebra Appl., 129:13-28 (1990). R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, N.J., 1962. Received 27 December 1990; final
nzanuscript
of nonnegative
matrices
accepted 9 July 1991
Appl.
and certain
Englewood
Cliffs,